Let S d be a unit sphere in ℝd+1, and let α be a positive real number. For pairwise different points x 1,x 2, . . . ,x N ∈ S d, we consider a functional E α (x 1,x 2, . . . ,x N ) = Σ i≠j ||x i − x j ||−α. The following theorem is proved: for α ≥ d − 2, the functional E α (x 1,x 2, . . . ,x N ) does not have local maxima.
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 65, No. 10, pp. 1427–1429, October, 2013.
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Radchenko, D.V. Local Maxima of the Potential Energy on Spheres. Ukr Math J 65, 1585–1587 (2014). https://doi.org/10.1007/s11253-014-0880-4
- Potential Energy
- Local Maximum
- Point Theorem
- Unit Sphere
- Positive Real Number